Resource Lesson
Continuous Charge Distributions: Electric Potential
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We have already derived in an earlier lesson that the electric potential at a distance r from a point charge is given by the formula
We will now construct a method to calculate the electric potential resulting from a continuous charge distribution in a similar fashion to how we developed expressions for the electric field due similar distributions.
Construct a diagram with a coordinate set of axes
Locate the point at which we want to calculate the absolute potential and label appropriate distances
Divide the total charge distribution into small charge segments (deltaq)
Develop an expression for one piece and then sum up the contributions for all of the charge segments
Replace the sum of the charge segments with an integral incorporating expressions for the charge density and an appropriate infinitesimal (ds, dA, dV).
Integrate and simplify
We will use this technique to calculate the electric potential along the axes of a thin ring of charge, a uniformly charged disk, and a concentric set of cylinders.
Uniformly Charged Ring
Suppose we have a ring of charge with a uniform charge distribution, λ, and radius a. We will now develop an expression for the electric potential at a position on the positive x-axis, at point P in the following diagram.
Since the ring has a uniform distribution of charge, we know that the total charge equals
allowing us to write an expression for the electric potential at point as
If we let x = 0, then we find that the potential at the center of the ring equals
Refer to the following information for the next three questions.
Suppose you have a 5 µC uniformly-charged ring of radius 10 cm whose origin is coincident with the origin. A small particle of mass 4 x 10
^{-6}
kg, having a charge of 2µC, is placed on the x-axis (the axis of the ring) at a distance of 20 cm.
What is the voltage of the ring at the particle's position?
What is the particle's initial potential energy?
When the particle is released, what will be its final velocity when it is a "great distance" from the ring?
Uniformly Charged Disks
To find the potential of a thin, charged disk having a uniform surface charge σ and radius R, start by placing its axis along the x-axis and build the disk from a series of charged rings.
Each ring will have a surface area of its circumference multiplied by its thickness, or
where the radius of the selected ring is a and its radial thickness is da. The charge carried on this charge segment would equal
where
With this initial setup, we can use our result for the potential of a thin ring to integrate from a radius of a = 0 to a = R to calculate the voltage due to the entire disk.
To integrate, use substitution where
and
this will now give us our conclusion:
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